1. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Chapter 2 Equations, Inequalities, and Problem Solving

2. 2-2 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Problem Solving The Five Steps for Problem Solving Percent Increase and Percent Decrease 2.5

3. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Five Steps for Problem Solving in Algebra 1. Familiarize yourself with the problem. 2. Translate to mathematical language. (This often means writing an equation.) 3. Carry out some mathematical manipulation. (This often means solving an equation.) 4. Check your possible answer in the original problem. 5. State the answer clearly, using a complete English sentence.

4. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. To Become Familiar with a Problem 1. Read the problem carefully. Try to visualize the problem. 2. Reread the problem, perhaps aloud. Make sure you understand all important words. 3. List the information given and the question(s) to be answered. Choose a variable (or variables) to represent the unknown and specify what the variable represents. For example, let L = length in centimeters, d = distance in miles, and so on. 4.Look for similarities between the problem and other problems you have already solved.

5. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. To Become Familiar with a Problem (continued) 5.Find more information. Look up a formula in a book, at the library, or online. Consult a reference librarian or an expert in the field. 6.Make a table that uses all the information you have available. Look for patterns that may help in the translation. 7.Make a drawing and label it with known and unknown information, using specific units if given. 8.Think of a possible answer and check the guess. Note the manner in which the guess is checked.

6. 2-6 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example The apartments in Wanda’s apartment house are consecutively numbered on each floor. The sum of her number and her next door neighbor’s number is 723. What are the two numbers? Solution 1. Familiarize. The apartment numbers are consecutive integers. Let x = Wanda’s apartment Let x + 1 = neighbor’s apartment

7. 2-7 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example 2. Translate. Rewording: Translating: 3. Carry out. x + (x + 1) = 723 2x + 1 = 723 2x = 722 x = 361

8. 2-8 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example If x is 361, then x + 1 is 362. 4. Check. Our possible answers are 361 and 362. These are consecutive integers and the sum is 723. 5. State. The apartment numbers are 361 and 362.

9. 2-9 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Digicon prints digital photos for $0.12 each plus $3.29 shipping and handling. Your weekly budget for the school yearbook is $22.00. How many prints can you have made if you have $22.00? Solution 1. Familiarize. Suppose the yearbook staff takes 220 digital photos. Then the cost to print them would be the shipping charge plus $0.12 times 220. $3.29 + $0.12(220) which is $29.69. Our guess of 220 is too large, but we have familiarized ourselves with the way in which the calculation is made.

10. 2-10 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example 2. Translate. Rewording: Translating: 3. Carry out.

11. 2-11 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example 4. Check. Check in the original problem. $3.29 + 155(0.12) = $21.89, which is less than $22.00. 5. State. The yearbook staff can have 155 photos printed per week.

12. 2-12 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example You are constructing a triangular kite. The second angle of the kite is three times as large as the first. The third angle is 10 degrees more than the first. Find the measure of each angle. Solution 1. Familiarize. Make a drawing and write in the given information. 2. Translate. To translate, we need to recall that the sum of the measures of the angles in a triangle is 180 degrees. 3x3x x x + 10

13. 2-13 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example 2. Translate (continued). 3. Carry out.

14. 2-14 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example The measures for the angles appear to be: first angle: x = 34 second angle: 3x = 3(34) = 102; third angle: x + 10 = 34 + 10 = 44 4. Check. Consider 34, 102 and 44 degrees. The sum of these numbers is 180 degrees and the second angle is three times the first angle. The third angle is 10 degrees more than the first. These numbers check. 5. State. The measures of the angles are 34, 44 and 102 degrees.

15. 2-15 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Percent Increase and Percent Decrease Whenever a percent of a quantity is calculated and then added to or subtracted from that quantity, the result is a percent increase or a percent decrease. Percent IncreasePercent Decrease Sales taxReduction in crimes Salary increaseLowered sale price Population growthBody weight loss

16. 2-16 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. There are three amounts in percent increase or decrease problems. 1. The original amount. 2. The amount of increase or decrease. 3. The new amount.

17. 2-17 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example After he had recovered from a prolonged illness, Victor’s weight increased from 140 lb to 150 lb. What was the percent increase in his body weight? Solution Identify the three amounts: Original Weight: 140 Amount of weight gained: 10 lb New Weight: 150 lb

18. 2-18 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example The amount of is what percent of original weight gained weight? 10=p140 The percent of increase is 7.1%.